By Garrett P.
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A 57 (1954), 218-230. [G] H. Garland, The arithmetic theory of loop groups, Publ. Math . lnst . Hauies Etud . Sci. 52 (1980), So136. [Jl] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloquium Pub!. I. (1968) . [J2] N. Jacobson, Exceptional Lie Algebras, M. Dekker Lect . Notes in Pure and Appl. Math . 1 New York (1971). L. Kantor, Classification of irreducible transitively differential groups, D ok!. Akad. Nauk SSSR 158 (1964), 1271-1274. [Ko] M. Koecher, Imbedding of Jordan algebras into Lie algebras, Amer.
26 7495 ON CONSTRUCTIONS OF NONSOLVABLE LIE ALGEBRAS WHOSE IDEALS ARE IN CHAIN M. PILAR BENITO Departamento de Matenuiticas y Computaci6n, Univers idad de La Rioja, Edificio de Magisterio y Matemciicas, Luis de Ulloa sin , 26004 Logrono, Spain Abstract. The principal task in this paper is giving some explicit constructions of nonsolvable Lie algebras -over fields of characteristic zero- in which the ideals are a n-element chain. Two different procedures are used in order to get the constructions: the first one depends on the radical of the Lie algebra and the second on the semisimple Levi factor.
1) . W~ a re i nterest~ d- i n 28 M. PILAR BENITO determining the products 0 from V(n) x V(n) into V(m) which satisfy (a,(3 E F ; u, v, wE V(n) and a E 81(2, F)) : i) (au+(3v)0w=a(u 0w)+(3(v 0w), ii) v 0 (au + (3w) = a(v 0 u) + (3(v 0 w), iii) a ยท (v 0 u) = (a . v) 0 u + v 0 (a . u). 1. Let V(n) , V(m) , sl(2, F) and 0 as it is described in the preceding paragraph. Then : 1) Ifm rt {2j : 0::; j ::; n} , 0 is the trivial product. 2) Ifm E {2j : 0 ::; j ::; n}, there exists a E F such that v0 0v n = aWm/2 ' Moreover, for each a E F and m there exists a unique product 0 0' satisfying i}, ii) and iii) which is completely determined by Vo 00' Vn = aWm/2 ' 3) If 0 is nontrivial, v 0 u = -v 0 u if and only if m is of the form 2n - 2(1 + 2j) where 0 ::; 1 + 2j ::; n; otherwise, v 0 u = u C:) v.
Some facts about discrete series (holomorphic, quaternionic) (2004)(en)(4s) by Garrett P.
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